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THE INFLUENCE OF RESPONSE FEEDBACK LOOPS ON THE LATERAL-DIRECTIONAL DYNAMICS OF A VARIABLE-STABILITY TRANSPORT AIRCRAFT

by Kenneth J. Szalai

Flight Research Center Edwards, Cali$

NATIONAL AERONAUTICS A N D SPACE ADMINISTRATION WASHINGTON, D. C. MAY 1967

https://ntrs.nasa.gov/search.jsp?R=19670017216 2020-03-02T09:33:42+00:00Z

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NASA TN D-3966

THE INFLUENCE OF RESPONSE FEEDBACK LOOPS ON THE

LATERAL-DIRECTIONAL DYNAMICS OF A

VARIABLE-STABILITY TRANSPORT AIRCRAFT

By Kenneth J. Szalai

Flight Research Center Edwards, Calif.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00

THE INFLUENCE OF RESPONSE FEEDBACK LOOPS ON THE

LATERAL-DIRECTIONAL DYNAMICS OF A

VARIABLE -STABILITY TRANSPORT AIRCRAFT

By Kenneth J. Szalai Flight Research Center

SUMMARY

Several response feedback loops are analyzed to determine their effects on the lateral-directional dynamics of a variable -stability transport aircraft. The response feedback system feeds back response variables such as sideslip angle o r roll rate as rudder o r aileron commands, or both, thus altering the various transfer functions which describe the dynamic characteristics of the aircraft. The range of feedback gain for which approximate expressions are valid in describing the effect of a particular loop is noted. The root-locus method is used to show how the Dutch roll, roll, and spiral modes a r e influenced as a function of feedback gain. Expressions are developed which directly relate feedback gains to some response parameter, such as Dutch roll frequency, damping ratio, o r roll and spiral mode time constants. The expansion to multiloop systems is discussed, along with limitations of the response feedback system, from the standpoint of operating a variable -stability aircraft.

INTRODUCTION

In conjunction with the design and development of a general purpose airborne simu- latorl (GPAS), studies were made at the NASA Flight Research Center of response feedback control loops and their influence on the GPAS, a variable-stability Lockheed JetStar aircraft. Although much information has been published on variable -stability aircraft, no single document contains a development of the fundamental concepts of response feedback systems along with a detailed discussion of loops that are utilized in variable-stability aircraft. Reference 1 is concerned primarily with showing the fea- sibility of installing a response feedback system (RFS) in a T-33 aircraft. The theoretical treatment is limited to showing that the aerodynamic coefficients of the equations of motion of the T-33 can be altered to match the coefficients of an aircraft to be simu- lated. The report does not analyze specific loops in detail, nor does it predict how individual loops influence the dynamic modes of the aircraft. It is not an adequate source of basic information on analysis techniques. Reference 2 is typical of many reports which emphasize research results of using an RFS aircraft but do not present

'NASA Contract NAS4-607 with the Cornel1 Aeronautical Laboratory, h c . , Buffalo, N. Y..

much background material on the theoretical basis of the system. This particular reference does contain several approximations derived from the equations of motion which are useful in describing the effects of certain RFS loops.

A qualitative description of a response feedback system is given in reference 3 , which discusses airborne simulation in general and its applications in research and training. Reference 4 is devoted entirely to one loop configuration and its effect on the spiral mode. It does contain some expressions that are useful in the description of the effect of several loops on the spiral mode. Reference 5 correlates root-locus analysis with the equivalent stability-derivative approach in the description of single-sensor control loops. The discussion is limited to two specific feedback loops and is not primarily concerned with variable-stability aircraft applications.

This paper summarizes the results of an analysis of the influence of response feedback loops on the lateral-directional dynamics of a variable-stability transport aircraft. It is intended to fi l l a significant gap in the available literature by presenting fundamental information on the response feedback system, demonstrating some of the more successful analysis techniques , and discussing in detail the feedback loops commonly used in variable-stability aircraft. Somewhat unique is the range of feedback gain for the JetStar for which certain classical approximations are valid in describing the effect of a particular loop.

SYMBOLS

The units used for physical quantities in this paper are given, where applicable, in both the International System of Units (SI) and U. S. Customary Units. relating the two systems are included in reference 6.

Factors

The body-axis system and sign conventions used in this paper are shown in figure 1.

AY BY C,QE coefficients of fourth-order algebraic equation

b wing span, m (ft)

CD

CZ

Cn

drag coefficient

rolling-moment coef - fic ient

yawing-moment coef - ficient

x +cp side -force coefficient

Xl

Figure 1.- Body-axis system and sign conventions.

2

acv

3

Ixz

natural log base

acceleration of gravity, m/sec2 (ft/sec2)

moments of inertia referred to X, Y, and Z body axes,

product of inertia referred to X and Z body axes, kg-m2 (slug-ft2)

2 respectively, kg-m (slug-ft2)

complex variable operator

designation of imaginary axis of s-plane, rad/sec (deg/sec)

a constant

artificially created stability derivative; rolling acceleration due to bank angle Lq

4

m mass, kg (slugs)

.

q S b b N P = I z z 2v c n j

artificially created stability derivative; yawing acceleration due to bank angle N9

P rolling angular velocity, rad /sec (deg/sec)

pitching angular velocity, rad/sec (deg/sec)

U

V

V

2 , N/m2 (lb/ft2) zpv dynamic pressure,

yawing angular velocity, rad/sec (deg/sec)

wing area, m2 (a2) Laplace transform variable

velocity component along X body axis, m/sec (ft/sec)

true airspeed, m/sec (ft/sec)

velocity component along Y body axis, m/sec (ft/sec)

W velocity component along Z body axis, m/sec (ft/sec)

5

x, y, z body -axis coordinates

CY angle of attack, rad (deg)

P angle of sideslip, positive for velocity vector toward right, rad (de@

6a total aileron deflection, Galeft - 6arighty positive for left aileron deflected trailing edge down, rad (deg)

pilot aileron command, positive when commanding positive 6,, rad (de&

6, rudder deflection, positive for trailing edge left, rad (deg)

command to rudder actuator o r aircraft transfer function, rad (deg) 6%

pilot rudder command, positive when commanding positive 6,, rad (de!%)

damping ratio of second-order dynamic response Cn

2 (s ) transfer -function nume rator parameter b 6a

Dutch roll damping ratio % 9

P

‘r

pitch-attitude angle, rad (deg)

mass density of air , kg/m3 (slugs/ft3)

designation of real axis of s-plane

roll mode time constant, sec

T S spiral mode time constant, sec

40 roll angle, rad (deg)

6

VA actuator phase lag, rad (deg)

arbitrary phase angle, rad (deg) * A

# heading angle, rad (deg)

w frequency , rad/sec

Wn undamped natural frequency of a second-order dynamic response ,

* ( s ) transfer -function numerator parameter

rad/sec

WV 6a

Dutch roll undamped natural frequency, rad/sec W#

Feedback loop and transfer-function nomenclature':

- 6a P

feedback loop which commands aileron proportional to roll rate, and

the actual value of feedback gain (e. g. , - 6a = 5 rad/rad/sec), rad/rad/sec (deg/deg/sec) P

feedback loop which commands aileron proportional to yaw rate, and a gain value, rad/rad/sec (deg/deg/sec)

feedback loop which commands aileron proportional to angle of side- sl ip, and a gain value, rad/rad (deg/deg)

feedback loop which commands Zilemn proportional to roll angle and a gain value, rad/rad (deg/deg)

feedback loop which commands rudder proportional to yaw rate, and a gain value , rad/rad/sec (deg/deg/sec)

feedback loop which commands rudder proportional to angle of side- - % P slip, and a gain value, rad/rad (deg/deg)

feedback loop which commands rudder proportional to time derivative - 6,

P of sideslip angle , and a gain value , rad /rad/sec (deg/deg/sec)

'Values of feedback loop gains and transfer function ratios in this paper are the same in both the SI and U. S. Customary Units.

7

Subscript:

actuator transfer function

roll rate to aileron transfer function

magnitude of roll angle to sideslip-angle transfer function evaluated at the Dutch roll root

handling-qualities parameter

yaw rate to rudder transfer function

T trim value

A dot over a quantity indicates the differentiation with respect to time; a double dot indicates the second differentiation with respect to time.

Special notation:

basic relates quantity to unaugmented aircraft, no feedback gains

closed loop with at least one feedback gain f O

effective including the augmentation of response feedback loops

open loop all feedback gains = 0, which results in a basic configuration

GENERAL DISCUSSION

In many cases the response feedback loops effectively augment the aerodynamic stability derivatives of the aircraft, hence the influence of these loops can be described by considering the usual effect of those particular stability derivatives on aircraft dynamics. However, some loops create aerodynamic stability derivatives that are nonexistent for the basic aircraft. The effect of these loops can be interpreted by noting the effect of the created stability derivatives. Where the classical approximations are not valid, root-locus diagrams and analog-computer results are used to describe the action of the loops on the airplane responses. These other methods are necessary when feedback gains a re high enough to augment some aerodynamic stability derivatives which, because of their usual small contribution to a certain dynamic mode, are not considered in the usual approximations.

8

Since the response feedback system can often be described in terms of equivalent stability derivatives, the effects of these loops vary somewhat, depending on the characteristics of the aircraft being considered. obtained for the GPAS from analytical studies of a Lockheed JetStar, an executive-type jet transport with aft-fuselage mounted engines. References to loop gain values should be indicative of values for other types of high-performance jet transports. Most large transports exhibit similar lateral-directional dynamics, which indicates similar transfer-function characteristics, and small variations in the basic aircraft transfer functions generally do not alter the dominant influence of a particular feedback loop. Thus, a particular loop described for the JetStar will have much the same effect on other aircraft in its class with some variation in the amount of feedback gain required to achieve certain closed-loop dynamics.

Since the individual feedback loops utilize the conventional aircraft control surfaces, the effectiveness of the loop is also a function of dynamic pressure and moments of inertia. Specific gains mentioned are valid only for a particular flight condition. Numerous references are made to flight condition, thus a code is used for convenience. Three parameters are written together in the following order: Mach number, weight, altitude (in thousands of meters (thousands of feet)). Thus, 0.55H6.1 denotes a Mach number of 0.55, heavy (17,236 kg (38,000 lb)), and 6100 meters (20,000 feet) altitude, and 0.23LO denotes a Mach number of 0.23, light (10,886 kg (24,000 lb)), at sea level.

The particular feedback loops discussed herein were chosen because they are the most effective for varying aircraft dynamics and are used on other variable-stability aircraft. Most root-locus diagrams presented are for the 0.23LO flight condition. The analysis was originally made for this case because it represented a low-dynamic- pressure condition, and hence gave information on minimum loop effectiveness. There is no other significance to the selection of this flight condition. for this case are representative of those throughout the flight envelope. Where significant differences do occur, another root-locus diagram o r special comment is included.

The quantitative results in this paper were

Root-locus diagrams

The response feedback loops analyzed are listed in the following table along with their estimated maximum gains, primary influence, and reasons for reaching a limit. These maximums were anticipated by the Cornel1 Aeronautical Laboratory on the basis of their experience with variable-stability aircraft and were used to predict the simulation capability of the GPAS. Actual limits can be determined only by flight test.

Primary Feedback derivative Estimated maximum

loop changed range of gain pr imary influence Reasons for reaching estimated

maximum

9

ANALYSIS OF SINGLE CONTROL LOOPS

In the following sections, two assumptions are made concerning the feedback loops '

that a r e analyzed:

1. The response sensors have no dynamics of their own. They yield a signal proportional to the response with no time lag.

2 . The actuators are also considered to be ideal. Actuator dynamics a re neglected.

These assumptions a re made in order to simplify the analysis to the point where basic loop effects are not clouded by hardware-imposed restrictions.

Each feedback loop can be represented as shown in figure 2 , in which the 6, - r

loop is used as an example. The commanded rudder 6, is a combination of the pilot

-7 I I I-ATrcraft -1 I function I

I I I

I

I I I

I

--------- I e(~) t ransfer function

I I t ransfer I

I rp

- r I

----- I I

- r I i

input 6 and the commanded by 'P

the feedback loop

6, Note that the feedback gain - can be r either positive o r negative. The rudder deflection, then, is a function of the aircraft's yaw-rate response. The aircraft transfer function is a mathematical model used to describe the input-output relationship between an aircraft control-

r l e r and an aircraft response. The - (s) transfer function is calculated from the Figure 2.- Feedback-loop configuration. 6, equations of motion (appendix A). stitution of equation (1) into the equations

Sub-

of motion will eliminate 6, and add three new terms to the equations:

(+) Ygrr, augmenting Yrr

(F) L6rr, augmenting Lrr

(F) N g r r , augmenting N r r

10

r The -(s) transfer function can be calculated from these modified equations and 6 r P 6, - will be a function of the feedback gain r. The feedback loop will not affect all

three terms equally but will depend on the relative magnitudes of Y6,, ‘6,’ 6,

r r and N6,. If, for example, N6, >> L6, and N6 >> Y6,, the - feedback loop

would act almost entirely as an N r augmentation loop.

also be in terms of 6 The equations will

instead of 6,. rP 6r r

loop does affect the equations of motion and the -(s)

transfer function will also be affected. It might be asked, then, which of the other transfer functions are changed, and, since the transfer functions a re usually presented in the form of one polynomial in s divided by another polynomial in s , whether the numerator o r denominator o r both will be a function of the feedback gain. nominators of all lateral-directional transfer functions are identical. when set equal to zero, is referred to as the characteristic equation. istic equation is altered by any feedback loop as shown in references 7 and 8. The effect of feedback loops on transfer-function numerators is not obvious (see, for example, ref. 9) , but the following facts are noteworthy and can be easily proved by deriving particular transfer functions which fall into the categories mentioned:

It is apparent that the 6 rP

The de- This polynomial, The character-

1. The closed-loop transfer function resulting from feeding back a response variable to one control input has the same numerator as the open-loop transfer

r 6, function. In other words, the numerator of the -(s) transfer function with a - r feedback loop is identical to the numerator of the -(s) transfer function with no

feedbacks, which is , of course, the open-loop transfer function.

6 r P r

6 r P

2, Loops which feed back some response variable to a particular control input will not alter the numerators of transfer functions relating any other response

6r variable to that same control input. That is, the r feedback loop will not alter

the &(s) or *(s ) transfer function numerators. 6t 6 r 3. It is possible to alter the numerator of a particular transfer function by

feeding back a different response variable to a different control input. Thus, the

numerator of the &(s) transfer function wi l l be altered by the r feedback loop.

r feedback loop. It will not be altered by the - feedback loop or the 6a

6r 6a

6r

This paper draws heavily from the interpretation of dynamic stability from s-plane considerations. These relationships are discussed in references 7 and 8 and in appendix B, which summarizes the application of root-locus results to dynamic stability analysis.

11

6r r The - Feedback Loop

6, r The - feedback loop, o r yaw damper, is common in many aircraft. One con-

venient method of analyzing this loop is to consider the two-degree-of-freedom approximation to the Dutch roll. motion in appendix A, the characteristic equation can be written approximately, as derived in appendix C , a s

By including the feedback equation in the equations of *

s 2 - [ N r + YP+ (!$) N~,]s + N p + N - NpY6,) = 0 (2)

Comparing equation (2) with the standard form of the second-order approximation to the Dutch roll, s2 + 2 g+ w + s + u+2 = 0 , results in

6r It is apparent from equations ( 3 ) and (4) that the r loop will affect both the

natural frequency and the damping ratio of the Dutch roll with varying degrees of effectiveness. For any given aircraft, numbers can be substituted into equations (3)

and (4) to find how effective r is in changing C + and w+. values of %, F ) N 6 . is of the same magnitude as N r + YP for the JetStar at most

flight conditions. Hence, for gain values of 7 between 1 and 2 .5 rad/rad/sec (deg/deg/sec), the 2C$w+ term is greatly changed. however, the (%) (N6,YP - NpY6,) term is small compared to NP + NrYp. There- fore, little variation is to be expected for small - gain values; thus, in

equation (3)

noting that N >>N,YP for the JetStar as well as most transport aircraft, equation (2) becomes

6r For relatively small

6r r

For the same range of gain,

6 r r

will be primarily affected. By neglecting the u+ variation and

P

s2 - INr + Y p + (“9 N6, I s + NP = 0 (5)

6, Thus, the - loop is almost entirely a Dutch roll damping loop for relatively small r gain values, on the basis of the classical two-degree-of-freedom Dutch roll approximation.

12

The root-locus technique can be used to relate the dynamic characteristics of the 6r aircraft directly to the value of the feedback gain -. Figure 3 is typical of most Jet- r

Star flight conditions. The locus shown is for a positive yaw rate fed back as a *

J -j2

s, Figure 3.- Root locus for the 7 loop at 0.23LO. (Gain values in rad/rad/sec.)

6, positive rudder input. The pole locations for r = 0 represent the open-loop aircraft.

For r < 2 . 5 rad/rad/sec, the Dutch roll damping ratio is affected, with little change

in u+, as the previous discussion predicted. The Dutch roll approximation does not

adequately describe the effect of the feedback loop for gtxins greater than

2.5 rad/rad/sec. Equation (5), then, is valid only for r < 2.5 rad/rad/sec. Above

this value of gain the Dutch roll damping ratio is still increased, but a marked increase in frequency is also evident. Hardware limitations (table, page 9) are expected to restrict the maximum feedback gain to about i4 rad/rad/sec for the JetStar, so for moderate gain values the feedback loop can be considered as a damping loop for the Dutch roll. the feedback loop had a pure effect on the Dutch roll damping ratio.

6 r

6r

The dashed line in figure 3 shows how the root-locus diagram would look if

6, The - feedback loop has some other noteworthy effects on the aircraft dynamics. r As the Dutch roll becomes more stable, the s p i r a l mode becomes more stable. (For a few flight conditions, 0 .75L20 , for instance, the zero locations are such that spiral

6r divergence results. ) The roll and spiral modes merge for - = 3 . 7 rad/rad/sec, r and for greater gain values a lateral phugoid made exists. It should be noted that this mode may not be attainable in practice, since a gain of 3 . 7 rad/rad/sec is near the expected maximum.

13

The root locus for the opposite sign of the feedback gain is shown in figure 4. Since root-locus diagrams are symmetrical with respect to the real axis, only the top

Dutch -0.5

1,1

4 Figure 4.- Root locus for the - 7 loop at 0.23LO. (Gain values in rad/rad/sec.)

Spiral :" r

:" r - -jl J -jl

6, Figure 5.- Root locus for the 7 loop at 0.55H6.1

(0.55H20). (Gain values in rad/rad/sec.)

occurs for a gain of 4 rad/rad/sec, which is the expected maximum gain.

6 r One of the practical problems in using the - loop occurs during a steady turn. 6 r For the stabilizing sense of the gain, the - loop commands a rudder deflection to r

oppose any yaw rate. In a turn this means that an increased amount of rudder-pedal input by the pilot is required to maintain some steady yaw rate. The usual method of

eliminating this problem is by inserting a llwashoutlv circuit in the 7 feedback loop, 6r

half is shown in this and succeeding diagrams. For very low feedback gain levels the spiral mode becomes divergent. As the gain level is increased, the h t c h roll becomes divergent as a result of the decrease in damping ratio to zero. The roll mode is virtually unaffected by the feedback loop in this case. If this loop is to be used to decrease Dutch roll damping, some means must be provided to maintain spiral stability.

The root locus for a higher dynamic pressure flight condition, 0.55H6.1 (0.55H20), is presented in figure 5. The dynamic pressure at this flight condition is 9911 N/m2 (206 lb/ft2), compared with 3735 N/m2 (78 lb/ft2) at 0.23LO. The roll mode time constant is larger, hence the pole is nearer the imaginary axis than before. The Dutch roll locus is nearly a perfect quadrant of a circle, which means

that for an increased r 6r gain an increase in CJ) can be realized with virtually no

change in Dutch roll natural frequency for g ~ ) between 0 and 1. The maximum

value of gain for which no change in WJI

occurs is 2 . 1 rad/rad/sec, approximately the value for which W J ) started to be

affected at the 0.23LO flight condition, even though the root locus is slightly different. The first breakaway into a lateral phugoid occurs with an attainable feedback gain, 2.3 rad/rad/sec. However, the high damping ratio (gn > 0.9) makes it difficult to detect. The second breakaway

14

which reduces the effectiveness of the loop at low frequencies. Hence, during a steady turn, the yaw-rate feedback gain is zero.

6r B The - Feedback Loop

For certain wings-level maneuvers, b can be assumed to be approximately equal 6r to -r. It would seem, then, that the effect of the -c- loop is similar in its effect

to that of the i r loop. Comparison of the root-locus diagrams in figures 3 and 6

shows the similarity. An expression can be derived relating the feedback T to a

Dutch roll characteristic and to aerodynamic stability derivatives of the basic aircraft. This relationship is shown in the following equation:

6 r b 6 r P

The derivation of this equation and the conditions of validity (appendix D) illustrate a useful approach which results in similar expressions. Equation (6) allows the gain to be calculated that will yield the value of 2C+w4 desired.

The root-locus plot of figure 6 is for the feedback configuration commanding a 6r negative 6, for a positive B . The figure shows that the - loop alters the Dutch roll B

damping ratio with little effect on w$, hence in equation (6) the selection of some

2 c + w + implies the choice of a particular

j w

Dutch roll 1 j2

J-j1

4 B

Figure 6.- Root locus for the -- loop at 0.23L0.

(Gain values in rad/rad/sec.)

with w remaining nearly constant at its

open-loop value. Equation (6) assumes that w+ does not vary, and figure 6

indicates that this assumption is good for E1 < 2. 5 rad/rad/sec. Over a fairly

large range of gain where 5 is varying rapidly (E I between 0 and 2 rad/rad/sec the roll and spiral modes are not influ-

enced as much as when - is used.

+

r 4

6 r r

If f l is obtained by differentiating p as measured by a vane, the loop may be noisy, thus limiting the feedback gain to relatively small values. Gains of much more than 1 rad/rad/sec are not to

be expected with a B signal derived by differentiation; hence, equation (6) is applicable for all practical gain values. To

15

circumvent the noisy 6 vane signal, yaw rate, and lateral acceleration and solving a simplified side-force equation (ref. 10). The resulting feedback gain to be used.

may be synthesized by measuring bank angle,

will be less susceptible to gust disturbances and will allow a higher

Reversing the sign of the feedback gain results in the root-locus diagram in figure 7. The 011 and spiral modes a re still unaffected. The Dutch roll becomes

divergent for 5 = 0.3 rad/rad/sec. The Dutch roll damping ratio is aga'n altered P 8r 6r with little change in u+. Comparison of the root-locus diagrams of the - and -

loops (figs. 2 , 3 , 5, and 6) shows the latter to be more nearly a pure Dutch roll damping loop, in that the roll and spiral modes a re not influenced as significantly as

P r

6, they are with the - feedback r jw loop.

'r The Y- loop adds the fol-

lowing three terms to the equations of motion:

P

L -3

'r - loop at 0.23LO. B Figure 7.- Root locus for the


These terms may be thought of as augmenting the three aerodynamic stability derivatives Yb, Lp, and NB. These stability derivatives do not appear in the equations of

motion for the JetStar (appendix A) because of their extremely small contribution. 'I- The -T- loop, then, can be regarded as actually creating the three derivatives, P although they do exist for the basic JetStar. In perturbation-type maneuvers, these

three derivatives have effects on lateral-directional dynamics similar to those of Yr, Lr, and Nr.

'r The - Feedback Loop P

6 r - P Angle of sideslip is detected and fed back as a rudder command to form the

loop. The two-degree-of-freedom approximation to the Dutch roll is derived (appendix C) by writing an auxiliary equation similar to equation (1) and then solving for the characteristic equation as follows:

16

For small values of

@)N6r, on the other hand, is comparable to NP, hence the Dutch roll frequency will

be primarily affected. When - is of the proper sign to increase w the total system

damping 25+u+ will increase. The two-degree-of-freedom approximation (eq. (7)) is

generally good for flight conditions of the JetStar where t r im angle of attack is small. If this is true, the following expression can be written relating the Dutch roll frequency

and the feedback gain -:

term is small compared to Nr + Yp. The term

6r P +'

6, P

The NP approximation for m+' is not satisfactory for large t r im angles of attack. It is necessary to use a more complete expression for w CYTLP, since this term is significant for large aT.

lesser degree, through the terms ?(NgS, $(Ygr), and !$(LG~), respectively.

which includes the term + 6, P

Note that the - loop acts as an effective NP, also augmenting Y p and Lp to a

j a

- j3

-1.5

Dutch roll

Roll Lo A I

-2 7 -3

Figure 8.- Root locus for the -- 'r loop at 0.23L0.

(Gain values in rad/rad.) P

A typical root-locus diagram for the JetStar is shown in figure 8. The locus represents the configuration in which positive p commands negative 6,. The

roll and spiral modes are altered very little. There is some slight sp i ra l destabilization, but even for very high gain the closed-loop root can be no farther into the right half of the complex plane than the open-loop zero. For the JetStar, this means that the time to double amplitude for the divergent term is no less than 40 seconds for any flight condi-

6r tion when the -- loop is being P used alone.

The effect on the Dutch roll is of primary importance, and the poles move in the manner predicted analytically, following a line of

17

nearly constant Z+W$. Since the Dutch roll mode poles a re

this class of aircraft, the small increase in C$W+ caused

9 P

damped for

results in the locus following a line of constant c+. The Dutch roll undamped natural frequency

6r is shown as a function of - in figure 9. fairly linear for

Above this gain

less effective in

lt

0 -4 -8 -12 -16

5 gain, rad/rad P

Figure 9.- Variation of 0 4 with the - 'r feedback P gain at 0.23LO.

Figure 10.- Root locus for the - 4 loop at 0.23LO.

(Gain values in rad/rad.) P

increasik U J I in the higher gain

ranges. Practical considerations restrict 1$1 to a maximum value not

much greater than 10 rad/rad. The plot indicates that the effort required to increase the upper limit of the gain from that in table I might not be justified in light of the small increase realized in u+.

gain results in the root locus of figure 10. The Dutch roll frequency is reduced, and at the same time the damping ratio decreases. For 1$1> 2 . 2 rad/rad the Dutch roll is unstable. The spiral mode is stabilized. The roll and sp i ra l modes merge and form an oscillatory mode. This mode may not be possible to attain in practice because of Dutch roll instability .

Reversing the sign of the feedback

It has been noted from figure 8 6, P

that the - loop would change the

total damping of the system very little. Figure 10 indicates that 2!!4w+ decreases rapidly and goes to zero for a fairly low value of feedback gain. This indicates that the two -degree -of - freedom approximation is no longer

a valid representation of the Dutch roll. From equation (7) it is noted that

18

By substituting actual values for the stability derivatives, it is found that for 6r - = 2.2 rad/rad, C p q , = P

validity for the 25-$w$ approximation. For large

by equation (9) is not valid.

The root locus, however, shows neutral stability.

* this gain area the effective ratio is high. Reference 5 discusses conditions of In

ratios the approximation given '

6r P $ The primary influence of - on the aircraft dynamics, that of changing o alone,

prevails for the stabilizing sense of the gain. works through the aerodynamic control derivatives, the effectiveness of the loops is a function of dynamic pressure. This dependency is shown by writing the expression for ~ $ 2 as

Since the response feedback system

gain, w $ ~ is directly propor- 6r It is evident from this equation that, for a given - P

tional to dynamic pressure for regions of the flight envelope where Cn and C

are constant. The ratio between the closed loop o$2 and the open loop o+2,

however, will be constant in this region for a given

q r P

6r - gain value, as shown by forming the ratio P

Cnp +&) CnBr 2 closedloop - -

0 L

open loop

6a The - Feedback Loop P

This feedback loop senses roll rate and commands aileron deflection propor- tionately. By writing an auxiliary equation similar to equation (1) and substituting it into the equations of motion (appendix A), it is noted that two new terms appear:

@) N6,p, augmenting NPp

For the transport class of aircraft, L6, is usually very much greater than Nga so loop influences the effective Lp almost entirely. The aerodynamic sta- 6a that the -

bility derivative Lp is strongly related to the roll mode time constant. In fact, the P

19

usual approximation, which is good for all JetStar flight conditions, is

6a Hence, the - loop exercises strong and direct control over the roll mode time P 6a constant. The - gain value must be relatively large before Np augmentation is P

significant, compared with the strong influence on the effective L through L6,. P

The root-locus diagram for the -- 6a loop is shown in figure 11. The

P spiral mode remains unchanged, and even high gains do not noticeably modify Dutch roll dynamics. The roll mode is gtabilized. By using equation (12), the O a - gain required for a particular roll

mode time constant can be calculated to be

-3

1

4 Figure 11.- Root locus for the -- loop at 0.23LO. P


l4 r

Exact solution-, 1 2 t 10 I-

The accuracy of equation (13) is shown in figure 12. The solid line represents the roll mode time constants resulting

from a variation of - obtained from

an exact solution to the equations of motion for the 0.55H6.1 (0.55H20) flight condition. The dashed line represents the roll mode time constants

6a and associated - gains obtained from P

equation (13) which assumes only Lp

6a P

augmentation. The close agreement shows the almost sole dependency of T r on Lp and indicates that equation (13) is a good approximation for

Equation (13) solution (Lp variation only)

2 'V' the JetStar.

0 -1 -2 -3 -4 -5 6a For a positive - gain, the root - 6 , gain, rad/rad/sec P P locus of figure 13 results. The roll

tion (13) solution. Figure 12.- Comparison of exact solution with eqna- mode is now destabilized. The roll and

spiral modes form a lateral phugoid as

20

- Dutch roll G-6a

-

lent to writing the open loop * ( s ) transfer function 6a

- increases and eventually become un- P

stable. The Dutch roll frequency is affected slightly, but the damping ratio is decreased noticeably.

j 2

- - - 0.5 P jl

For roll-mode analysis of the Jet-

(14) Figure 13.- Root locus for the loop at 0.23LO. P


and noting that for the JetStar the two second-order terms are close to cancellation.

This fact can be interpreted in terms of the handling-qualities ratio ?!!P . For the

JetStar and similarly configured transport aircraft, this ratio is approximately 1. The complex pole and zero will be ve ry close to each other, depending on how close the

ratio &? is to 1. Thus, the Dutch roll poles will usually close on the complex zeros,

not influencing the locus on the real axis.

u+

crc,

6, The - Feedback Loop P

This feedback loop commands aileron deflection proportional to angle of sideslip. The aerod-fnamic stability derivatives Lp and NP will be modified through the

fol lo~ing new terms:

(2) Lgap, augmenting Lpp

For the JetStar and similar aircraft, L6, is usually much greater in magnitude than

N6,. hence, L augmentation will dominate, The influence of this loop varies more

from one flight condition to another than the loops previously discussed but generally has a strong influence on the spiral mode and a noticeable effect on the Dutch roll mode. The action of this loop in providing a rolling moment due to sideslip is recognized as the dihedral effect. When L is increased negatively, the spiral mode is generally

stabilized and the Dutch roll mode destabilized.

P

P

21

- stable for - gains between -10 and

j l -20 rad/rad. The frequency at which this occurs may be larger o r smaller than the basic Dutch roll frequency.

P

-3 -2 -1 1 6a (9911 N/m2 (207 lb/ft2)), where - P increases the Dutch roll frequency. For

Dutch roll

P -

- j2 The exact shape of the Dutch roll locus again is subject to flight condition, and for some higher dynamic pressures the Dutch roll frequency increases for the posi- jl

-3 -2 -1 1 2 (table, pg. 9), the Dutch roll dynamics a r e definitely affected, but the - 6a loop is not nearly as

6a The r Feedback Loop

6a The - loop commands aileron deflection proportional to yaw rate. It is not one r . of the primary control loops that would be used alone but could find applications in

multiloop combinations o r in special cases. By augmenting Nr and L,, this loop

- I.,

would be expected to alter Dutch roll damping.

lJ1 -4 1 4 % 7 - = 0 Spiral

s a Figure 16.- Root locus for the loop at 0.23LO.


i w

Dutch roll 4 jl

A representative root -locus diagram is shown in figure 16. For the Jetstar, the complex zeros are generally in the right half of the plane for all but high dynamic pressure flight conditions. Relatively small values of feedback gain will cause Dutch roll instability. The spiral mode is stabilized as the effective L r is decreased.

Reversing the sign of the feedback results in the root locus of figure 17. The spiral mode is divergent for very small gain values, usually less than 0.5 rad/rad/sec for most flight conditions. The spiral instability occurs for much smaller gain values than does Dutch roll instability, thus making use of this loop alone to alter Dutch roll effects impractical.

6a The - FeedbackLoop ‘p

6a The - loop, commanding

aileron proportional to bank angle, is somewhat distinctive in that it creates two aerodynamic stability derivatives that are nonexistent for the basic aircraft:

v

e ’ N g , p , creating N ‘p ‘p

23

The root locus for the -- loop is shown in figure 18. The pole-zero 40 6a configuration of figure 18 is identical to that of the -- root locus (fig. 11) except that -

P there is no zero at the origin. The spiral mode is stabilized and the roll mode de-

stabilized. Dutch roll dynamics a re af- jw fected very little. This loop can be used

to create a low-frequency mode commonly - referred to as the lateral phugoid. In- creasing the feedback gain causes the roll and spiral modes to merge and form an oscillatory pair. The gain at breakaway varies with flight condition and is

j 2 Dutch roll

6, G - = - 1 . 5 1 cp - j l

i -0 .3 rad/rad for the 0.23LO condition. \I I . I ,, Spiral , (T This loop is useful in itself in stabilizing

Roll I

I \ - - the spiral mode o r in conjunction with other loops in stabilizing the spiral mode without interfering with the Dutch roll

-3 -2 1

- _ 6a ~ - 0 . 3 CF dynamics. - -jl

'a Relating the spiral pole location to the feedback gain is not easily done because the spiral root cannot be simply expressed in terms of stability derivatives. When

Figure 18.- Root locus for the - - loop at 0.23LO.

(Gain values in rad/rad.)

the characteristic equation of the system is written a s

As4 + Bs3 + Cs2 + Ds + E = 0 (15)

it is possible to approximate the spiral root in terms of the coefficients of equation (15). If the assumption is made that a single-order root exists that is much smaller than any of the other roots, a good approximation to this root is

By writing the complete characteristic equation from the set of lateral-directional equations in appendix A, including the auxiliary feedback equation, in the form of equation (E), the 5 term can be formed as E

The derivation of equation (17) is presented in appendix E.

feedback loops,

For the aircraft without any

is usually very small. The balance between LpN, and LrNp

24

determines the sign of the fraction and, hence, determines if the spiral root is con- vergent o r divergent. Investigation of relative magnitudes of terms in the numerator

6a and denominator indicates that a very small variation of - is required to control cp

* the spiral mode. 6a In appendix E, the Nq terms, (7)Nfja, were much smaller than the Lq terms,

6 ($)Ld,. and so were neglected. Thus, the 3 loop acts as though it creates an v alone. However, it is important to note that this does not imply that an equivalent Ncp

will not affect the spiral mode. If Nq can be created where Lq is insignificant, such 6r as with a - loop o r with wing-tip drag surfaces, spiral stability will be greatly cp

affected, as shown in reference 4.

L c p

Roll

-3 5 2 -1 1 .l I - I.,

oa q

The - loop induces the derivative Nq with the conventional aileron control surface, through N6,, which is a very small number for the JetStar. This feedback

loop can also be used to destabilize the spiral mode by reversing the sign of the feedback gain. The root locus for this case is shown in figure 19. The roll mode is noy

isfactory results as long as

interesting to note that all of the

TS ,Spiral , is not very large. It is also -

aerodynamic stability derivatives augmented by previously discussed hops appear in cqua-

1 F

"a stabilized, and the value of cp for which the spiral mode is divergent is small, usually less than 0 . 0 5 rad/rad.

The expression relating spiral root location will give sat-

1

MULTILOOP RESPONSE FEEDBACK OPERATION

The response feedback loops have thus far been considered a s single-loop systems. In most practical applications, however, it is necessary for two o r more loops to be used simultaneously to achieve the desired aircraft response characteristics. If, for example, it is required that both Dutch roll frequency and damping ratio be controlled, it is necessary to use more than one feedback loop, since no one loop can xercise independent control over both of these parameters. It was noted that the - loop

affects the Dutch roll frequency and the

gr 6r P

loop has its major effect on the Dutch roll

25

damping ratio, hence the two loops can be used together to adjust the system frequency and damping ratio. The block diagram of this system is shown in figure 20. By re-

arranging the diagram, it is possible ~~ to write a +s) transfer function

6 rP for the entire closed-loop system as . I B a s f c r Z a f t l

I

I

I I P

r I I = - -4s) I r 6 r

+ + ‘re I - I

I

l ( 8 ) = [@‘!basic (18) I 6’P 1 - (“1.1 r I6, L(’) -0 [es)]basic ,

‘r, - 0 t = 4 s ) .

I The characteristic equation, which is the denominator of equation (18) set equal to zero, is a function of both feedback gains , hence two variables exist which influence the movement of poles on the s-plane.

6 1’ L-,,-

Figure 20.- Two-loop block diagram representation.

The root-locus technique can be used to analyze certain multiloop systems as shown in reference 8. The result is a root contour, as shown in figure 21. The

6r 6, dashed lines are constant - gains; the solid lines represent constant values of P r -. ‘r

0 - r

2 . 5

-8

-6 - ‘r 0’ I

p - 4 H ’Dutch

roll / -2 / ./’

// / Roll

I 1 * I Y

-3 - -1 I

6r ---- Lines of constant - P Lines of constant r 6r

i 4

i 3

i 2

Spiral , a 1

-j 1

and $ loops at 0.23LO. Figure 21.- Root contour for the y sr s, sr

respectively .) P (Gain values for y and - in rad/rad/sec and rad/rad,

26

By using such a plot, it is possible to achieve a desired W+ and C+. The roll 6r 6r .and spiral modes are not shown in the figure, but each combination of - and - P r

will determine a roll and a spiral time constant. If a somewhat independent means of . varying T~ is required, it will be necessary to add another loop, - 6a , for example.

P Spiral stability may also be important; hence, a fourth loop would be necessary. Thus, there are many combinations of loops which could be used, and these configurations are dictated by the requirements on the closed-loop response.

The method for analyzing a multiloop feedback system involving a single control

input, 6ap o r 6, can be summarized as follows: P'

1. Close one loop and find the root locus corresponding to various values of feedback gain.

2. For one value of the feedback gain of loop number one, locate the roots (poles) and add the zeros of loop number two, disregarding the original zeros.

3. Find the root locus corresponding to the pole-zero configuration of step 2, as a function of the second feedback gain.

Note that step 3 may be repeated for every value of the first feedback gain desired, in each case using the pole locations corresponding to that gain. The resulting diagram is a root contour, showing graphically the effect of two feedback loops. The process can be expanded to more loops only if the loops have the same control-surface input.

6 r 6a This method is not valid for a r and a - loop combination, for example, inasmuch P as the implicit assumption of this technique is that closing the first loop did not alter the zeros of the second loop in figure 20. statements made earlier concerning the effects of feedback loops on transfer-function numerators.

This assumption follows from the general

For multiple input-output systems , the characteristic equation must be written for the total system and factored to yield the roots which determine the transient response of the aircraft. This must be done for each combination of feedback gains. Reference 9 presents a more detailed analysis of general multiloop systems.

In some cases, it is possible to develop an expression relating two feedback gains 6, 6r

and a dynamic mode variable. Equation (19) applies when the - and T loops are used together as P P

27

Equation (19) was obtained by examining equation (6) and noting that - Y6, augments

y which results from the 6r loop used alone.

6r (3 and indicates the required T gain for a desired 25+w+. The u+ now is that

P P 3-

In most cases it is not possible to write simple expressions relating two or more feedback gains, and it is necessary to resort to factoring the total system characteristic equation, given the feedback gain values, to find the closed-loop dynamics.

ANALYSIS OF TWO-LOOP OPERATION FROM STABILITY-DERIVATIVE CONSIDERATIONS ALONE

In some cases, the use of feedback loops to alter the dynamics of an aircraft is not conveniently analyzed by root-locus methods. Although quantities such as frequency, damping, and time constants are easily picked off s-plane root-locus plots, other

parameters often used in handling-qualities studies are not, such as 2 P or pq+ - Often, these parameters a re of primary concern in a response feedback loop application. Analysis of such applications is possible by working with the equivalent aerodynamic stability derivatives, that is, the effective stability derivatives, resulting from feedback loop augmentation.

13s ) I * In many handling-qualities studies, it is desirable to vary the ratio

Reference 2 contains the following simplified expression for this ratio when Q is small :

Any feedback loop which augments LP or N

6r 6r tion (20). Rewriting the equation to include the effects of a - loop (where - P P carries its own sign) introduces new terms as shown in the following expression:

will affect the - (s) ratio of equa- P 1; I*

Changes inthe I;(s)l+ ratio are necessarily accompanied by changes in the Dutch roll natural frequency because of the NB augmentation. The use of - alone is likewise

undesirable because of its influence on the effective NP through the term &)N6,, as shown by the following equation:

6a P

28

the effect of both the - 6r and the p 6a

and yet maintain the effective NP P

. By using an expression for

loops, it is possible to

constant, assuming

is small due to a small aT. This expression is to u~i', that is , the a L T P

If the total effective NP is to remain constant at the open-loop value, the relation

N6a '- P (68a)Ng, _ - - -

must be satisfied. Substituting equation (24) into equation (23) results in the expression

The desired (s) ratio can be obtained by solving for the - gain in equation (251,

then solving for the corresponding - gain from equation (24). The absolute-value

sign indicates that there will be two sets of gains which will satisfy equation (25). The 6a selection of the sign of the - gain to be used must be made in the light of other con- P

siderations, such as the fact that Lp is an important handling-qualities parameter in

itself, and for conventional aircraft in the transport category has a minus sign associated with it, which indicates positive dihedral. This means that the sign of the 6a - gain must be chosen that will maintain the total effective LP as a negative quantity. P

la I$ 6 r P P

Other dynamic side effects of the two-loop combination used here, such as the influence on the spiral mode, may dictate the use of other loops to maintain a flyable vehicle. The effect of these other loops on the IF(s)lJi ratio, in turn, may have to be

taken into account. Iterative processes will usually be necessary to determine the final

29

combination of gains, and these processes are best handled by a digital computer. some cases when equation (20) is valid, equation (25) may be adequate to calculate the required gains. In the preceding example, it was assumed that it was necessary to hold N constant at the open-loop value, but equation (24) could just have easily been P written to attain some other desired effective Np. In practice, there will be many cases in which equation (20) will not be adequate. This must be determined first. The

simplified approximation to I z(s)) was used to illustrate the equivalent stability-

derivative approach in a two-loop situation.

For

.

.

p +

EFFECT OF ACTUATOR DYNAMICS ON RFS PERFORMANCE

Analysis of the response feedback loops has been carried out on an idealized system where neither sensors nor actuators display any phase lag. The scope of this paper does not warrant a detailed examination of all the associated effects of sensor and servo dynamics; however, it is worthwhile at this point to consider how these effects can be included in the analysis of a typical feedback loop.

The problem of phase lags in a response feedback system is a serious one. Generally, sensors, such as gyros, accelerometers, and vanes, can be expected to have less phase lag than the actuators at operating frequencies. The effect of actuator dynamics on response -feedback-system performance is, then, of primary interest.

A suitable mathematical model must be chosen to represent the actuator. One such model is shown in figure 22. The actuator can be shown in the aircraft response

feedback system in the forward loop as in figure 23. The closed-loop transfer function can then be written as

wn I

Figure 22.- Mathematical model of a second-order actuator.

2 Aircraf t t ransfer 6% Kw n function +

- I s2+ 2cnwns+ un2 &(s )

P 6,

Figure 23.- Inclusion of actuator in RFS loop.

P -

30

The order of the characteristic equation is increased from a fourth- to a sixth-order equation. The phase lag of the actuator itself can be calculated at any driving frequency w as

It is apparent from equation (27) that in order to minimize phase lag wn should be as large as possible for a given En.

The addition of an actuator in the forward loop does not greatly increase the labor required for analysis by the root-locus method. The poles (and zeros, if any) of the actuator transfer function are placed on the s-plane with the poles and zeros of the &(s) transfer function. The location of the actuator poles is a function of 5, and on. 6, It is usually desirable to have approximately 70 percent critical dam ing for the

actuators. The effect of such a 70-percent damped actuator on the - loop and its influence on the Dutch roll are shown in figure 24.

& P

Natural frequency of actuator, rad/sec

No actuator dynamics

r a d h a d

J - j l

Figure 24.- Effect of actuator on Dutch roll dynamics with

'r

B the - - loop closed at 0.23LO.

The locus for the system when actuator dynamics are neglected was discussed in the section on the 6 r - loop. The 63 rad/sec actuator P has destabilized the Dutch roll by decreasing the damping ratio. For an actuator with a natural frequency of 25 rad/sec, the Dutch roll can be made to go divergent if the gain is high enmgh, in this case, above 20 rad/rad. Actuators with natural frequencies less than 25 rad/sec would cause instability at lower values of feedback gain.

The roll and spiral modes will not be affected noticeably unless they form a lateral phugoid and break away from the real axis. The breakaway point itself will be dependent somewhat on actuator pole location. The representation of the actuator may be different from the ideal second-order type assumed here. Position rate o r pressure rate feedbacks around the actuator, along with other compen- sation, could result in a higher

31

order representation with possible numerator roots. In this case, the roll and spiral modes would be influenced. It is also important that the actuator poles be examined as the feedback gain is increased. While it is possible that these poles will move into the right-half plane, resulting in an unstable system, it is also likely that they may move toward the real axis, indicating that the damping ratio of the actuator is increasing. Equation (27) shows that the phase lag will increase with increasing En, and it is undesirable to add any more phase lag to the system.

*

The importance of including actuator dynamics in practical applications is evident from figure 24, and especially when high gain feedback loops are used with low frequency actuators.

CONCLUSIONS

On the basis of an analysis of the response feedback system for the lateral- directional modes of a variable-stability transport airplane , the following general conclusions can be drawn:

1. Each feedback loop influences all of the aircraft dynamic modes but, in most cases, displays one dominant effect.

2. For values of feedback gain that are considered to be reasonable for this airplane, most classical approximations of loop action were valid.

3. The analytically derived gain expressions have conditions of validity associated with them, and it is important to ascertain that these conditions a re satisfied.

4. The equivalent stability -derivative method of analysis is valuable when root- locus diagrams are not convenient to show loop effects.

5. Root-locus analysis techniques can be expanded to miltiloop configurations, but it must be realized that the methods outlined herein a re applicable only to loop combinations that do not alter the numerators of transfer functions in the closed loop.

6. Actuator dynamics can be readily included in a root-locus analysis. In more detailed studies of response feedback systems it will usually be necessary to include actuator effects.

7. Practical limitations will determine the final success of a response feedback system. These limitations must be included ,in the analysis as demanded by the particular application.

Flight Research Center, Nat iona l Aeronautics and Space Administration,

Edwards, Calif. , December 14, 1966 7 2 0 -0 4 -00 -0 1 -2 4

32

APPENDIX A

EQUATIONS OF MOTION USED IN LATERAL-DIRECTIONAL RFS STUDY

The side-force and moment equations (ref. 11) are written in a set of orthogonal body-axis coordinates. The three equations written in conventional nonlinear form are as follows:

Side force -

Rolling moment -

Yawing moment -

These three equations a re modified by assuming that, in still air,

W a x - V

p " " V

To further reduce complexity with little loss in accuracy, the following approximations are made:

1. Inertia coupling terms in the moment equations a re dropped

V 2. p = - V

3. -CD sin p + C y cos p is approximated by Cy

33

APPENDIX A

4. V is used for u in the side-force equation

The side-force equation then becomes

m(b+ r V - p w ) = m g s i n c p c o s e + p V 1 2 SCY

Dividing by mV,

$ + r - Q - v - g sin cp cos e + E c ~

Incorporating the approximations,

--c +g sin cpcos e+ pa! - r b - 2 m Y v

The rolling-moment equation is

The yawing-moment equation is

The three equations are then linearized about a t r im flight condition (V = VT) where the initial p, r, (9, $, p, 6r, and 6a are zero. The resultant linearized

perturbation equations are (assuming C y . = C = Cl . = 0) P ys, P

Side force -

Rolling moment -

34

APPENDIX A

Yawing moment -

As a further modification to conserve analog equipment necessary for programing, the Euler angular rate relationship is solved for p as

p = (s - J , s i n e

Substituting the Euler expression for 4, p = (p - (q sin 9 + r cos ‘p) tan e

Linearizing this expression and assuming tan eT = tan aT = aT,

Differentiating ,

Substituting p and p into the three equations and rearranging, using the dimensional forms of the aerodynamic stability derivatives, yields the final form of the lateral- directional equations of motion. Note that the equations are now in terms of an earth reference, @ and v, rather than a body axis, p and b, as follows:

Side force -

Rolling moment -

Yawing moment -

35

APPENDIX A

By Laplace transforming the equations and using the variable s , the equations can be written as

Side force -

Rolling moment -

Yawing moment -

36

APPENDIX B

APPLICATION OF ROOT-LOCUS METHODS TO ANALYSIS OF RESPONSE FEEDBACK LOOPS

The root-locus method of analysis (refs. 7 and 8) directly relates feedback gain values and characteristic equation roots. It is these roots that determine the form of the response of the dynamic system. Generally, the characteristic equation for the lateral-directional mode is a fourth-order equation which, for the transport class of aircraft, usually factors into the following form (ref. 5):

Each parenthetical term is referred to as a dynamic mode. The second-order term is the IXltch roll mode, and the two first-order terms are the roll and spiral modes, respectively. Adequate descriptions of the physical characteristics of these modes a re found in most texts on aircraft dynamics. For the analyses of this paper it is sufficient to note that each parenthetical term in the equation in the Laplace variable (frequency domain) has an equivalent form in the time domain, as explained in references 7 and 8. The Dutch roll is an exponentially varying sinusoidal oscillation of the general form

where h is an arbitrary phase angle. The roll and spiral modes are , respectively, t

exponential functions of the form e - F and e TS . The root-locus technique provides a coiwenier,t xethod of plotting the roots of the characteristic equation as a function of the particular feedback gain used. The resultant curve, presented on the s-plane, yields information about the dynamic stability of each of the modes. the three modes and indicates the parameters characteristic of the dynamics of each mode.

t --

Figure 25 shows

The basic aircraft characteristic equation roots are designated by X's and are referred to as poles. The Dutch roll mode appears as two poles because the roots of the second-order term in the characteristic equation a re a complex pair. The magnitude of the distance between the origin and one Dutch roll pole is the undamped natural frequency in radians/second. The component of this distance along the imaginary axis is the damped frequency that would be measured on a time history. The damping ratio of the Dutch roll is the cosine of the angle between the negative real axis and a line drawn from the origin to the pole. The roll and spiral mode time constants, in seconds, correspond to the inverse of the distance between the origin and the pole location. If a root lies to the left of the ju axis, the resultant time response is stable; whereas, roots falling to the right of the ju axis a re unstable. Roots falling on the ju axis are termed neutrally stable.

37

APPENDIX B

Graphical techniques used in plotting root-locus diagrams utilize the transfer function numerator roots (zeros) as well as the poles. These zero locations are designated by 0 ' s and are shown herein on root-locus diagrams. The graphical techniques are described in references 7 and 8.

Imaginary axis

Stable jw Unstable Dutch roll I (+)

I t

x Dutch rol l

Figure 25.- Lateral-directional modes as displayed on the complex o r s-plane.

38

APPENDIX C

TWO-DEGREE -0 F-FREEDOM APPROXIMATION TO

THE DUTCH ROLL WITH THE % LOOP

By choosing only the dominant terms in the equation of motion, the side-force and yawing-moment equations become

and

-Npp+ (s - Nr) r = N6,6,

allowing only p and r motions. NOW letting 6, = 6rp + M r ,

-N p + (s - Nr) r = Ng 6 + e ) N ( j r r P r rP o r

the characteristic equation is

S2 - [Nr + Yp + ($)N~Js + Np + NrYp + E)(Ng,Yp - NpY6,) = 0

In similar fashion, writing 6, = 6 to the Dutch roll can be used to obtain the characteristic equation

+ - p, the two-degree-of-freedom approximation rP (“p.1

39

APPENDIX D

6, B ANALYTICAL EXPRESSION RELATING - TO

OF THE DUTCH ROLL

The expanded characteristic equation general literal form is easily shown (ref. 5) to be

+ - 1 + -)s 1 3 Tr Ts

Comparing the coefficients of this equation with

E A A A s 4 + A s E 3 + - s c 2 + - s + - D = o

which is the general form of a fourth-order algebraic equation where the coefficient of the highest order term s4 has been made equal to unity, and equating the s3 terms of the two equations results in

1 For most transport aircraft, - can be neglected with respect to the other two terms. T S

Also, to a very good approximation

and so B

= 2 & p # - Lp

40

1

APPENDIX D

After writing the feedback loop equation as

. and substituting into the lateral-directional equations of motion, the characteristic equation can be found. The coefficient of s4 term (A) is

and the coefficient of s3 term (B) is

The s4 and s3 coefficients can be sim lified by neglecting small triple-product , small numbers themselves, appear in terms; inertia terms, since - IXz and -

products usually much smaller than other terms in the sum; and terms involving aT, since the t r im angle of attack is assumed to be small. Using the remaining significant terms, the first two terms of the characteristic equation are

pxz Ixx I z z

and

But, as shown previously,

= 2E$W$ - Lp A

41

APPENDIX D

and, substituting the expressions derived for A and B,

6r Equating the two expressions for and solving for 7 9

P

42

APPENDIX E

t

*

APPROXMATION TO THE SPIRAL MODE TIME CONSTANT WHEN 6a IS USED AS A FEEDBACK LOOP

4p

Reference 5 states that if the spiral mode constant T~ is very large,

where D and E are the coefficients of the s1 and so terms, respectively, of the lateral-directional characteristic equation. If the auxiliary feedback equation is written as

D, the coefficient of the s1 term, is

The E t e r m i s

43

APPENDIX E

?

Neglecting small triple-product terms and assuming that inertia terms are small,

since Ixz << 1 for the JetStar, ?

Iz z

For the JetStar,

and

D = LpNp - LpNp - & Lp +

Similarly for the E term, neglecting small triple-product terms and inertia terms,

For the JetStar,

Furthermore, for most cases,

>> LpNGa

E and so the fraction can be written as

l7 1

44

REFERENCES

<

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Ball, J. N. ; et al. : Installation of an Automatic Control System in a T-33 Airplane for Variable Stability Flight Research. Part 1 - Preliminary Investi- gation and Design Studies. Tech. Rep. 55-156 (Cornel1 Rep. No. TB-936-F-l), Wright A i r Dev. Center, U.S. A i r Force, Apr. 1955.

Kidd, Edwin A. ; and Harper, Robert P. , Jr. : Fixed-Base and In-Flight Simulations of Longitudinal and Lateral-Directional Handling Qualities for Piloted Reentry Vehicles. Tech. Doc. Rep. No. ASD-TDR-61-362, A i r Force Systems Command, Wright-Patterson A i r Force Base, Feb. 1964.

Kidd, Edwin A. ; Bull, Gifford; and Harper, Robert P., Jr. : In-Flight Simulation - Theory and Application. Rep, No. 368, AGARD, 1961.

Anon. : Flight Evaluations of the Effect of Variable Spiral Damping in a JTB-26B Airplane. Rep. No. TB-1094-F-1, Cornel1 Aero. Lab., Inc., Oct. 19, 1957.

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NASA-Langley, 1967 - 2 H-431 45

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